Valentin Féray ; Ekaterina A. Vassilieva - Linear coefficients of Kerov's polynomials: bijective proof and refinement of Zagier's result

dmtcs:2815 - Discrete Mathematics & Theoretical Computer Science, January 1, 2010, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) - https://doi.org/10.46298/dmtcs.2815
Linear coefficients of Kerov's polynomials: bijective proof and refinement of Zagier's resultArticle

Authors: Valentin Féray 1; Ekaterina A. Vassilieva 2

  • 1 Laboratoire Bordelais de Recherche en Informatique
  • 2 Laboratoire d'informatique de l'École polytechnique [Palaiseau]

We look at the number of permutations $\beta$ of $[N]$ with $m$ cycles such that $(1 2 \ldots N) \beta^{-1}$ is a long cycle. These numbers appear as coefficients of linear monomials in Kerov's and Stanley's character polynomials. D. Zagier, using algebraic methods, found an unexpected connection with Stirling numbers of size $N+1$. We present the first combinatorial proof of his result, introducing a new bijection between partitioned maps and thorn trees. Moreover, we obtain a finer result, which takes the type of the permutations into account.


Volume: DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)
Section: Proceedings
Published on: January 1, 2010
Imported on: January 31, 2017
Keywords: Kerov's Character Polynomials,Bicolored Maps,Long Cycle Factorization,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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